Question

In the figure above, AC is a diameter of the circle. If AC = 1, which of the following gives the area of triangle ABC in terms of ? B) A) 2 tan 2 C) 2 sine sin cos D)

Answer 1

Assuming that triangle ABC is a right triangle, then its area is computed as follows:

$A=\frac{AB\cdot BC}{2}$

From definition,

sin(angle) = opposite/hypotenuse

cos(angle) = adjacent/hypotenuse

Then,

$\begin{gathered} \sin \theta=\frac{AB}{AC} \\ \text{ Given that AC = 1} \\ \sin \theta=AB \end{gathered}$$\begin{gathered} \cos \theta=\frac{BC}{AC} \\ \text{Given that AC = 1} \\ \cos \theta=BC \end{gathered}$

Replacing these results into the area equation,

$A=\frac{\sin \theta\cdot\cos \theta}{2}$